# Experimentation at the Frontiers of Reality in Schubert Calculus

@article{Hillar2009ExperimentationAT, title={Experimentation at the Frontiers of Reality in Schubert Calculus}, author={Christopher J. Hillar and Luis David Garc{\'i}a-Puente and Abraham Mart{\'i}n del Campo and James Ruffo and Zach Teitler and Stephen L. Johnson and Frank Sottile}, journal={arXiv: Algebraic Geometry}, year={2009}, volume={517} }

We describe the setup, design, and execution of a computational experiment utilizing a supercomputer that is helping to formulate and test conjectures in the real Schubert calculus. Largely using machines in instruc- tional computer labs during off-hours and University breaks, it consumed in excess of 350 GigaHertz-years of computing in its first six months of operation, solving over 1.1 billion polynomial systems. This experiment can serve as a model for other large scale mathematical…

## 11 Citations

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A rich story concerning intrinsic structure, or Galois groups, of Schubert problems is also beginning to emerge from experimentation, which showcases new possibilities for the use of computers in mathematical research.

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We describe a large-scale computational experiment to study structure in the numbers of real solutions to osculating instances of Schubert problems. This investigation uncovered Schubert problems…

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The Mukhin-Tarasov-Varchenko Theorem (previously the Shapiro Conjecture) asserts that a Schubert problem has all solutions distinct and real if the Schubert varieties involved osculate a rational…

Frontiers of reality in Schubert calculus

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The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus…

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The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus…

The Monotone Secant Conjecture in the Real Schubert Calculus

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- 2015

The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real, and is a compelling generalization of the Shapiro conjecture for Grassmannians.

Solving schubert problems with Littlewood-Richardson homotopies

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A new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians based on Vakil's geometric proof of the Littlewood-Richardson rule is presented.

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This proposal is to support my scientific work and that of my advisees in the areas of combinatorial and real algebraic geometry, including related work in applications of algebraic geometry. This…

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We discuss the so-called secant conjecture in real algebraic geometry, and show that it follows from another interesting conjecture, about disconjugacy of vector spaces of real polynomials in one…

The Secant Conjecture in the Real Schubert Calculus

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We formulate the secant conjecture, which is a generalization of the Shapiro conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with…

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